Superfield Approach to Nilpotency and Absolute Anticommutativity of Conserved Charges: 2D Non-Abelian 1-Form Gauge Theory

Abstract

We exploit the theoretical strength of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism to express the nilpotency and absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST conserved charges for the two (1 + 1)-dimensional (2D) non-Abelian 1-form gauge theory (without any interaction with matter fields) in the language of superspace variables, their derivatives, and suitable superfields. In the proof of absolute anticommutativity property, we invoke the strength of Curci-Ferrari (CF) condition for the (anti-)BRST charges. No such outside condition/restriction is required in the proof of absolute anticommutativity of the (anti-)co-BRST conserved charges. The latter observation (as well as other observations) connected with (anti-)co-BRST symmetries and corresponding conserved charges are novel results of our present investigation. We also discuss the (anti-)BRST and (anti-)co-BRST symmetry invariance of the appropriate Lagrangian densities within the framework of AVSA. In addition, we dwell a bit on the derivation of the above fermionic (nilpotent) symmetries by applying the AVSA to BRST formalism, where only the (anti)chiral superfields are used.

Modified 2D Proca Theory: Revisited under BRST and (Anti-)chiral Superfield Formalisms

“…Equation (140)). We point out that the restriction B + B − 2 E = 0also appears in (141) which is, once again, similar to the observation in 2D non-Abelian theory in the context of the existence of the off-shell nilpotent and absolutely anticommuting (anti-)co-BRST symmetries [39].…”

“…We would like to point out here that both the factorized terms in Equation (141) are zero separately and independently because both of them owe their origins to mathematically independent cohomological operators of differential geometry. For instance, as pointed out earlier, the restriction (B + B − 2 E = 0) owes its origin to the exterior derivative We would like to mention a few things connected with the 2D non-Abelian 1-form gauge theory which we have discussed in our earlier work [39] where we have shown the existence of the (anti-)co-BRST symmetries (in addition to (anti-)BRST symmetries). In fact, we have derived the off-shell nilpotent and absolutely anticommuting (anti-)co-BRST symmetry transformation for the 2D non-Abelian 1-form gauge theory under which the Lagrangian densities and, specifically, the gauge-fixing term remain invariant.…”

“…This also gets reflected at the level of the conserved and nilpotent charges Q ðaÞb because the absolute anticommutativity of these charges (i.e., (Q b Q ab + Q ab Q b = 0)), once again, crucially depends on the existence of CF condition (B + B + ðC × CÞ = 0). In addition, we note that the applications of the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations s ðaÞb on the Lagrangian densities L B and L B of the non-Abelian 1-form gauge theory lead to the following (see, e.g., [39]):…”

Nilpotent symmetries of a 4D model of Hodge theory: Augmented (anti-)chiral superfield formalism

“…In this context, it is gratifying to mention that we have already applied this idea to 1D rigid rotor and 2D self-dual bosonic field theory and obtained the expected results [39,40]. In a very recent set of papers [41,42], we have exploited our present ideas and shown the validity of absolute anticommutativity of conserved charges for the non-Abelian 1-form gauge theory (without any interaction with matter fields) and interacting Abelian 1-form gauge theory with Dirac and complex scalar fields. It is also gratifying to state that our present theoretical technique has been applied to interacting non-Abelian 1-form gauge theory with Dirac fields and we have established the absolute anticommutativity of the conserved and off-shell nilpotent (anti-)BRST charges [43].…”

“…( 7)), and vice-versa. We have algebraically played with the expressions in the ordinary space and superspace which have helped each-other in the derivation of equations ( 38), (40), (41), (43), ( 45)-(47). In other words, we have been able to prove the nilpotency as well as absolute anticommutativity of the fermionic (anti-)BRST and (anti-)co-BRST charges due to our knowledge of superfield formalism and properties of the fermionic symmetry transformations in the ordinary space.…”